Three-Dimensional Geometry

Let $P$ be the plane such that it contains the straight line $\dfrac{x-1}2=\dfrac{y-3}3=\dfrac{z+2}1$ and is perpendicular to the plane $x+2y+3z=4$. Let $P_1$ be the plane which passes through the point $(4,2,2)$ and is parallel to $P$.

Then which of the following statements is (are) TRUE ?

Let $L$ be the straight line joining the points $P(1,2,-1)$ and $Q(2,3,1)$. Let $S$ be the foot of the perpendicular drawn from the point $R(4,-1,5)$ to the line $L$. Another line passing through $R$ intersects $L$ at a point $T$ such that the point $S$ divides the line segment $PT$ internally in the ratio $|PS|:|ST|=1:2$, where $|PS|$ and $|ST|$ are the lengths of the line segments $PS$ and $ST$, respectively.

Then which of the following statements is (are) TRUE ?

If the point of intersection of the lines $\dfrac{x+1}3=\dfrac{y+a}5=\dfrac{z+b+1}7$ and $\dfrac{x-2}1=\dfrac{y-b}4=\dfrac{z-2a}7$ lies on $xy$-plane, then the value of $a+b$ is :

Let a line $L$ passing through the point $(1,1,1)$ be perpendicular to both the vectors $2\hat i+2\hat j+\hat k$ and $\hat i+2\hat j+2\hat k$. If $P(a,b,c)$ is the foot of perpendicular from the origin on the line $L$, then the value of $34(a+b+c)$ is :

Let the point A be the foot of perpendicular drawn from the point $P(a,b,0)$ on the line $\dfrac{x-1}2=\dfrac{y-2}1=\dfrac{z-\alpha}3$. If the midpoint of the line segment PA is $\left(0,\dfrac34,\dfrac{-1}4\right)$, then the value of $a^2+b^2+\alpha^2$ is equal to :

A line with direction ratios $1,-1,2$ intersects the lines $\dfrac x2=\dfrac y3=\dfrac{z+1}3$ and $\dfrac{x+1}{-1}=\dfrac{y-2}1=\dfrac z4$ at the points P and Q, respectively. If the length of the line segment PQ is $\alpha$, then $225\alpha^2$ is equal to:

The square of the distance of the point $(-2,-8,6)$ from the line $\dfrac{x-1}1=\dfrac{y-1}2=\dfrac z{-1}$ along the line $\dfrac{x+5}1=\dfrac{y+5}{-1}=\dfrac z2$ is equal to:

The shortest distance between the lines $\vec r=\left(\dfrac13\hat i+2\hat j+\dfrac83\hat k\right)+\lambda\left(2\hat i-5\hat j+6\hat k\right)$ and $\vec r=\left(-\dfrac23\hat i-\dfrac13\hat k\right)+\mu\left(\hat j-\hat k\right)$, $\lambda,\mu\in\mathbb R$, is:

If $\left(2\alpha+1,\ \alpha^2-3\alpha,\ \dfrac{\alpha-1}2\right)$ is the image of $(\alpha,2\alpha,1)$ in the line $\dfrac{x-2}3=\dfrac{y-1}2=\dfrac z1$, then the possible value(s) of $\alpha$ is (are)

The square of the distance of the point $P(5,6,7)$ from the line $\dfrac{x-2}2=\dfrac{y-5}3=\dfrac{z-2}4$ is equal to:

The square of the distance of the point of intersection of the lines $\vec r=\left(\hat i+\hat j-\hat k\right)+\lambda\left(a\hat i-\hat j\right)$, $a\ne0$ and $\vec r=\left(4\hat i-\hat k\right)+\mu\left(2\hat i+a\hat k\right)$ from the origin is:

Let a triangle PQR be such that P and Q lie on the line $\dfrac{x+3}8=\dfrac{y-4}2=\dfrac{z+1}2$ and are at a distance of 6 units from $R(1,2,3)$. If $(\alpha,\beta,\gamma)$ is the centroid of $\Delta PQR$, then $\alpha+\beta+\gamma$ is equal to :

If the distance of the point $(a,2,5)$ from the image of the point $(1,2,7)$ in the line $\dfrac x1=\dfrac{y-1}1=\dfrac{z-2}2$ is 4, then the sum of all possible values of a is equal to :

Let the image of the point $P(1,6,a)$ in the line $L:\dfrac x1=\dfrac{y-1}2=\dfrac{z-a+1}b$, $b>0$, be $\left(\dfrac a3,0,a+c\right)$. If $S(\alpha,\beta,\gamma)$, $\alpha>0$, is the point on L such that the distance of S from the foot of perpendicular from the point P on L is $2\sqrt{14}$, then $\alpha+\beta+\gamma$ is equal to:

Let a line L be perpendicular to both the lines $L_1:\dfrac{x+1}3=\dfrac{y+3}5=\dfrac{z+5}7$ and $L_2:\dfrac{x-2}1=\dfrac{y-4}4=\dfrac{z-6}7$. If $\theta$ is the acute angle between the lines L and $L_3:\dfrac{x-\frac87}2=\dfrac{y-\frac47}1=\dfrac z2$, then $\tan\theta$ is equal to:

The shortest distance between the lines $\dfrac{x-4}1=\dfrac{y-3}2=\dfrac{z-2}{-3}$ and $\dfrac{x+2}2=\dfrac{y-6}4=\dfrac{z-5}{-5}$ is :

Let the image of the point $P(0,-5,0)$ in the line $\dfrac{x-1}2=\dfrac y1=\dfrac{z+1}{-2}$ be the point R and the image of the point $Q\left(0,\dfrac{-1}2,0\right)$ in the line $\dfrac{x-1}{-1}=\dfrac{y+9}4=\dfrac{z+1}1$ be the point S. Then the square of the area of the parallelogram PQRS is ________.

Let the foot of perpendicular from the point $(\lambda,2,3)$ on the line $\dfrac{x-4}1=\dfrac{y-9}2=\dfrac{z-5}1$ be the point $(1,\mu,2)$. Then the distance between the lines $\dfrac{x-1}2=\dfrac{y-2}3=\dfrac{z+4}6$ and $\dfrac{x-\lambda}2=\dfrac{y-\mu}3=\dfrac{z+5}6$ is equal to :

Let a line $L_1$ pass through the origin and be perpendicular to the lines
$L_2:\vec r=(3+t)\hat i+(2t-1)\hat j+(2t+4)\hat k$ and
$L_3:\vec r=(3+2s)\hat i+(3+2s)\hat j+(2+s)\hat k$, $t,s\in\mathbf R$.
If $(a,b,c)$, $a\in\mathbf Z$, is the point on $L_3$ at a distance of $\sqrt{17}$ from the point of intersection of $L_1$ and $L_2$, then $(a+b+c)^2$ is equal to ________.

Let $L_1$ be the line of intersection of the planes given by the equations

$$2x + 3y + z = 4 \quad\text{and}\quad x + 2y + z = 5.$$

Let $L_2$ be the line passing through the point $P(2, -1, 3)$ and parallel to $L_1$. Let $M$ denote the plane given by the equation

$$2x + y - 2z = 6.$$

Suppose that the line $L_2$ meets the plane $M$ at the point $Q$. Let $R$ be the foot of the perpendicular drawn from $P$ to the plane $M$.

Then which of the following statements is (are) TRUE?